Meanwhile, Djokovic's struggles continue as he departs yet another tournament early.
Evidently confident, the Yorkshireman broke twice in the first set and then crucially held off two break points at the start of the second to stay on even terms.
After breaking the two-time Madrid champion in the third, Edmund saved three break points before serving out the final set to progress.
Nadal remains on course after a comprehensive victory over Monfils. "But, also, he was courageous enough to attack the balls when it mattered and deserved to win".
"So that's why it was really a flawless scenario to start off the tournament". So, it's a really good win for me. The teenager got the better of his fellow Canadian Milos Raonic in a 6-4, 6-4 win.
The final quarter-final will be between second seed Alexander Zverev - a 6-4, 6-2 victor over Leonardo Mayer, and John Isner, who edged Pablo Cuevas 6-7 (9), 7-6 (3), 7-6 (4).
Sentimental support, too, for another veteran Spaniard, Fernando Verdasco, also a leftie but with a heavyweight baseline game that had taken him, at his best, to No7 in the world and through Nadal in a memorable five-set thriller at the 2016 Australian Open.
Kohlschreiber is set to play the third round against South Africa's Kevin Anderson, sixth-seed and a finalist at the 2017 US Open, who defeated Kazakhstan's Mikhail Kukushkin, 5-7, 7-6 (3), 6-2.
"It'll be another tough match", Wozniacki said. I hope he's obviously back, who knows about Wimbledon. It's just about going for it again.More news: James Murdoch Won't Move to Disney if Fox Deal Closes
More news: Red Sox, Yankees to meet in London
More news: Joe Burrow To Transfer From Ohio State
Alongside the brilliance, though, there were uncharacteristic mistakes, noticeably an unforgivable missed overhead in the second set that, fortunately for the Serbian, didn't come back to haunt him. "He [Djokovic] is a legend of the game, one of the best tennis players of all time". It's still important for people, well, myself, to realize, and I do.
Belarusian veteran Azarenka was featuring in her first tournament on clay for two years and came from 3-0 down to level in the final set against Pliskova.
Edmund was delighted to overcome another highly ranked player after beating world No.10 Goffin.
Nadal extended his winning streak to 50 consecutive sets on clay, eclipsing the mark McEnroe established on carpet in 1984.
With momentum on her side, Sharapova won 20 of the last 22 points to secure a spot in the last-16.
"Pleased with the way my game's developing, that I'm able to beat a player like that".
Novak Djokovic of Serbia in action during his match against Kei Nishikori of Japan during day three of the Mutua Madrid Open tennis tournament at the Caja Magica on May 7, 2018 in Madrid, Spain.
"Then I really set up a good opportunity for me to step up".